The foundations of rationality, as LW knows it, are not defined with logical rigour. Are you adamant this is not a problem?
http://lesswrong.com/lw/31/what_do_we_mean_by_rationality/ says:
We are not here to argue the meaning of a word, not even if that word is "rationality". The point of attaching sequences of letters to particular concepts is to let two people communicate - to help transport thoughts from one mind to another. You cannot change reality, or prove the thought, by manipulating which meanings go with which words.
I don't think it's very helpful to oppose a logical definition for a certain language that would allow you to do this. As it is, you currently have no logical definition. You have this:
Epistemic rationality: believing, and updating on evidence, so as to systematically improve the correspondence between your map and the territory. The art of obtaining beliefs that correspond to reality as closely as possible. This correspondence is commonly termed "truth" or "accuracy", and we're happy to call it that.
Instrumental rationality: achieving your values. Not necessarily "your values" in the sense of being selfish values or unshared values: "your values" means anything you care about. The art of choosing actions that steer the future toward outcomes ranked higher in your preferences. On LW we sometimes refer to this as "winning".
That is not a language with a formalized type system. If you oppose a formalized type system, even if it were for the advancement of your purely practical goal, why? Wikipedia says:
A type system associates a type with each computed value. By examining the flow of these values, a type system attempts to ensure or prove that no type errors can occur. The particular type system in question determines exactly what constitutes a type error, but in general the aim is to prevent operations expecting a certain kind of value from being used with values for which that operation does not make sense (logic errors); memory errors will also be prevented.
What in a type system is undesirable to you? The "snake oil that cures lung cancer" - I'm pretty sure you've heard about that one - is a value whose type is irrational. If you may use natural language to declare that value as irrational, why do you oppose using a type system for doing the same thing?
I've seen there's discussion on LW about rationality, namely, about what it means. I don't think a satisfactory answer can be found without defining what rationality is not. And this seems to be a problem. As far as I know, rationality on LW does not include systematic methods for categorizing and analyzing irrational things. Instead, the discussion seems to draw a circle around rationality. Everyone on LW is excepted to be inside this circle - think of it as a set in a Venn diagram. On the border of the circle there is a sign saying: "Here be dragons". And beyond the circle there is irrationality.
How can we differentiate the irrational from the rational, if we do not know what the irrational is?
But how can we approach the irrational, if we want to be rational?
It seems to me there is no way to give a satisfactory account of rationality from within rationality itself. If we presuppose rationality is the only way to attain justification, and then try to find justification for rationalism (the doctrine according to which we should strive for rationality), we are simply making a circular argument. We already presupposed rationalism before trying to find justification for doing so.
Therefore it seems to me we ought to make a metatheory of rationality in order to find out what is rational and what is irrational. The metatheory itself has to be as rational as possible. That would include having an analytically defined structure, which permits us to at least examine whether the metatheory is logically consistent or inconsistent. This would also allow us to also examine whether the metatheory is mathematically elegant, or whether the same thing could be expressed in a simpler form. The metatheory should also correspond with our actual observations so that we could figure out whether it contradicts empirical findings or not.
How much interest is there for such a metatheory?